A New Characterization of Besov Spaces on the Real Line

We give a new characterization of functions ƒ defined on the real line (−∞, ∞) in order to belong to a Besov space Bp,rα for some 0 < α < 1 and 1 ≤ p, r ≤ ∞. These conditions are in terms of the Riesz mean of ƒ in case 1 ≤ p ≤ ∞, and in terms of the Dirichlet integral of ƒ in case 1 < p < ∞. An analogous characterization of periodic functions on the torus [−π, π) was initiated by Fournier and Self, via the partial sums of their Fourier series. The novelty in our treatment is that we use norms involving integrals, instead of norms involving sums of infinite series. Our approach is also appropriate to building up a complete characterization of Besov spaces on the torus